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Perfect colourings of simplices and hypercubes in dimension four and five with few colours

Dirk Frettlöh

Abstract

A vertex colouring of some graph is called perfect if each vertex of colour $i$ has the same number $a_{ij}$ of neighbours of colour $j$. Here we determine all perfect colourings of the edge graphs of the hypercube in dimensions 4 and 5 by two and three colours, respectively. For comparison we list all perfect colourings of the edge graphs of the simplex in dimensions 4 and 5, respectively.

Perfect colourings of simplices and hypercubes in dimension four and five with few colours

Abstract

A vertex colouring of some graph is called perfect if each vertex of colour has the same number of neighbours of colour . Here we determine all perfect colourings of the edge graphs of the hypercube in dimensions 4 and 5 by two and three colours, respectively. For comparison we list all perfect colourings of the edge graphs of the simplex in dimensions 4 and 5, respectively.
Paper Structure (6 sections, 6 theorems, 8 equations, 3 figures, 2 tables)

This paper contains 6 sections, 6 theorems, 8 equations, 3 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Colour adjacency matrices of hypercubes and simplices
  4. All perfect colourings of the 4-simplex and the 5-simplex
  5. Perfect colourings of the 4-hypercube
  6. Perfect colourings of the 5-cube

Key Result

Theorem 2.2

Let $M$ be the adjacency matrix of some graph $G$ and let $A$ be the colour adjacency matrix of some perfect colouring of $G$. Then the characteristic polynomial of $A$ divides the characteristic polynomial of $M$. In particular, each eigenvalue of $A$ is an eigenvalue of $M$ (with multiplicities).

Figures (3)

  • Figure 1: A perfect colouring of the edge graph of the cube with three colours. The corresponding colour adjacency matrix is $( 012102111 )$.
  • Figure 2: The perfect colourings of the 4-cube with two and three colours.
  • Figure 3: A perfect 3-colouring of the 5-cube with colour adjacency matrix $\left(050104023\right)$. This is indeed the graph of the 5-cube if we imagine the image wrapped on a cylinder, so edges that leave to the left are connected to those that enter on the right on the same height.

Theorems & Definitions (10)

  • Definition 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Lemma 4.1
  • proof
  • Theorem 5.1
  • proof
  • Theorem 6.1
  • proof
  • Lemma 6.2